# Tangential approximation of analytic sets

## TL;DR

This paper introduces a refined notion of tangential s-equivalence for real subanalytic sets with isolated singularities and shows that high-order Taylor truncations of analytic maps preserve this tangential equivalence.

## Contribution

It defines tangential s-equivalence considering tangent planes and proves high-order Taylor truncations of analytic maps produce algebraic sets tangentially s-equivalent to the original.

## Key findings

- High-order Taylor truncations preserve tangential s-equivalence.
- Introduces a new notion of tangential s-equivalence for singular sets.
- Shows algebraic approximations can match analytic sets in tangential behavior.

## Abstract

Two subanalytic subsets of $ \mathbb R^n$ are called $s$-equivalent at a common point $P$ if the Hausdorff distance between their intersections with the sphere centered at $P$ of radius $r$ vanishes to order $>s$ as $r$ tends to $0$. In this work we strengthen this notion in the case of real subanalytic subsets of $\mathbb R^n$ with isolated singular points, introducing the notion of tangential $s$-equivalence at a common singular point which considers also the distance between the tangent planes to the sets near the point. We prove that, if $V(f)$ is the zero-set of an analytic map $f$ and if we assume that $V(f)$ has an isolated singularity, say at the origin $O$, then for any $s\geq 1$ the truncation of the Taylor series of $f$ of sufficiently high order defines an algebraic set with isolated singularity at $O$ which is tangentially $s$-equivalent to $V(f)$.